Sharpen mesh curves by dissolving vectors
In situations like this, Select Similar can save the day

As your intuition hinted to you, converting it to a mesh first (AltC) and dealing in vertices is the best approach.

(I'm including all keyboard shortcuts to be friendly to a general audience.)



Once that is done, in Mesh Edit Mode Select All with A and press W to Remove Doubles.

Then Extrude (E) upward on the Z axis (Z) so that you will have some face angles (I know, it's unfortunate that it doesn't work with edge angles). 

With one corner edge selected you can then select all the 90-degree angles using Select Similar (ShiftG).



Add these selected corner vertices to a Vertex Group so that you can re-select them later. (Click the "+" button to create a new group, then the "Assign" button)



You can now delete the vertices that were created during extrusion. Selecting them from the front or side orthographic views and use Border Select (B). then Delete (X).



Use the trusty Vertex Group you created to call up the corner vertices. (Click the "Select" button)

Then invert your selection using CtrlI to select everything that is not a corner. From there it's just a matter of hitting X and choosing Dissolve Vertices.



Here is the result:



That concludes the solution for right angled corners.



For any regions that are not right angles, you can select their vertices separately and Remove Doubles (W) with a high Merge Distance threshold.

(You may want to avoid selecting the outermost vertices, because if they are collapsed during the merge operation the angles of the walls they comprise would change.)



Merge Distance can be set in the Operator Panel, so make sure it's visible (you may need to press T and drag that window area up from the bottom if it's not already visible).



The result after removing doubles at an increased merge distance:



In the above example, if you needed to eliminate the very short edge segments that remain, you could perform an Edge Slide (GG), slide the selected edge all the way toward the longer walls, select all relevant vertices, and Remove Doubles (W).

Each use case will of course be a little different, but with the tools I've described and a bit of ingenuity you should be good to go.